Halacha seems to imply it is problematic to partake in these activities on Shabbos yet of late I have noticed it has become a Mitzvah to do it. What is the Halachic Basis for it? And which Posek talks about it? What does he say?
Since nowadays no one is going to make a musical instrument on the spot - which was the reason for the original Gezeira not to dance and clap - the Biur Halacha 339:3 and Tosfos, Beitzah 30a says the Nohagim Lehakail. The Aruch HaShulchan 339:9 says that the original Rabbinical decree applied only to dance movements which required musical accompaniment, not to the unstructured and informal circle dancing popular today. Notwithstanding the above clapping with a Shinui is permitted according to all opinions - see Orach Chaim 339:3; Mishnha Berurah 338:1.
maran forbids, rama mekel but for mitsva. in practive is forbidden for sefaradim and some ashkenazim do it with conditions.
see ialcut iosef where he brings that it is proper to refrain even to walk in circles like some people do it the synagogue, while it is not considered dancing because people doesn't lift the legs (so from icar hadin is permitted) he recommends not doing it.